Heaps, Heap Sort, and Priority Queues

Trees A tree T is a collection of nodes T can be empty (recursive definition) If not empty, a tree T consists of a (distinguished) node r (the root), and zero or more nonempty subtrees T1, T2, ...., Tk

Some Terminologies Child and Parent Leaves Sibling Every node except the root has one parent  A node can have an zero or more children Leaves Leaves are nodes with no children Sibling nodes with same parent

More Terminologies Path Length of a path Depth of a node A sequence of edges Length of a path number of edges on the path Depth of a node length of the unique path from the root to that node Height of a node length of the longest path from that node to a leaf all leaves are at height 0 The height of a tree = the height of the root = the depth of the deepest leaf Ancestor and descendant If there is a path from n1 to n2 n1 is an ancestor of n2, n2 is a descendant of n1 Proper ancestor and proper descendant

Example: UNIX Directory

Example: Expression Trees Leaves are operands (constants or variables) The internal nodes contain operators Will not be a binary tree if some operators are not binary

Background: Binary Trees root Has a root at the topmost level Each node has zero, one or two children A node that has no child is called a leaf For a node x, we denote the left child, right child and the parent of x as left(x), right(x) and parent(x), respectively. Parent(x) x leaf leaf left(x) right(x) leaf

A binary tree can be naturally implemented by pointers. Struct Node { double element; // the data Node* left; // left child Node* right; // right child // Node* parent; // parent } class Tree { public: Tree(); // constructor Tree(const Tree& t); ~Tree(); // destructor bool empty() const; double root(); // decomposition (access functions) Tree& left(); Tree& right(); // Tree& parent(double x); // … update … void insert(const double x); // compose x into a tree void remove(const double x); // decompose x from a tree private: Node* root; A binary tree can be naturally implemented by pointers.

Height (Depth) of a Binary Tree The number of edges on the longest path from the root to a leaf. Height = 4

Background: Complete Binary Trees A complete binary tree is the tree Where a node can have 0 (for the leaves) or 2 children and All leaves are at the same depth No. of nodes and height A complete binary tree with N nodes has height O(logN) A complete binary tree with height d has, in total, 2d+1-1 nodes height no. of nodes 1 1 2 2 4 3 8 d 2d

Proof: O(logN) Height Proof: a complete binary tree with N nodes has height of O(logN) Prove by induction that number of nodes at depth d is 2d Total number of nodes of a complete binary tree of depth d is 1 + 2 + 4 +…… 2d = 2d+1 - 1 Thus 2d+1 - 1 = N d = log(N+1)-1 = O(logN) Side notes: the largest depth of a binary tree of N nodes is O(N)

(Binary) Heap Heaps are “almost complete binary trees” All levels are full except possibly the lowest level If the lowest level is not full, then nodes must be packed to the left Pack to the left

1 4 2 5 2 5 4 3 6 1 3 6 A heap Not a heap Heap-order property: the value at each node is less than or equal to the values at both its descendants --- Min Heap It is easy (both conceptually and practically) to perform insert and deleteMin in heap if the heap-order property is maintained

Structure properties Has 2h to 2h+1-1 nodes with height h The structure is so regular, it can be represented in an array and no links are necessary !!! Use of binary heap is so common for priority queue implemen-tations, thus the word heap is usually assumed to be the implementation of the data structure

Heap Properties Heap supports the following operations efficiently Insert in O(logN) time Locate the current minimum in O(1) time Delete the current minimum in O(log N) time

Array Implementation of Binary Heap C A B C D E F G H I J 1 2 3 4 5 6 7 8 … D E F G H I J For any element in array position i The left child is in position 2i The right child is in position 2i+1 The parent is in position floor(i/2) A possible problem: an estimate of the maximum heap size is required in advance (but normally we can resize if needed) Note: we will draw the heaps as trees, with the implication that an actual implementation will use simple arrays Side notes: it’s not wise to store normal binary trees in arrays, because it may generate many holes

class Heap { public: Heap(); // constructor Heap(const Heap& t); ~Heap(); // destructor bool empty() const; double root(); // access functions Heap& left(); Heap& right(); Heap& parent(double x); // … update … void insert(const double x); // compose x into a heap void deleteMin(); // decompose x from a heap private: double* array; int array-size; int heap-size; }

Insertion Algorithm Add the new element to the next available position at the lowest level Restore the min-heap property if violated General strategy is percolate up (or bubble up): if the parent of the element is larger than the element, then interchange the parent and child. 1 1 1 2 5 2 5 2 2.5 swap 4 3 6 4 3 6 2.5 4 3 6 5 Insert 2.5 Percolate up to maintain the heap property

Insertion Complexity Time Complexity = O(height) = O(logN) A heap! 7 9 8 17 16 14 10 20 18 A heap! Time Complexity = O(height) = O(logN)

deleteMin: First Attempt Algorithm Delete the root. Compare the two children of the root Make the lesser of the two the root. An empty spot is created. Bring the lesser of the two children of the empty spot to the empty spot. A new empty spot is created. Continue

Example for First Attempt 1 2 5 4 3 6 2 5 4 3 6 2 5 4 3 6 1 3 5 4 6 Heap property is preserved, but completeness is not preserved!

deleteMin Copy the last number to the root (i.e. overwrite the minimum element stored there) Restore the min-heap property by percolate down (or bubble down)

An Implementation Trick (see Weiss book) Implementation of percolation in the insert routine by performing repeated swaps: 3 assignment statements for a swap. 3d assignments if an element is percolated up d levels An enhancement: Hole digging with d+1 assignments (avoiding swapping!) 7 7 7 4 9 8 9 4 8 8 17 16 14 10 17 14 10 17 9 14 10 4 20 18 20 18 16 20 18 16 Dig a hole Compare 4 with 16 Compare 4 with 9 Compare 4 with 7

Insertion PseudoCode void insert(const Comparable &x) { //resize the array if needed if (currentSize == array.size()-1 array.resize(array.size()*2) //percolate up int hole = ++currentSize; for (; hole>1 && x<array[hole/2]; hole/=2) array[hole] = array[hole/2]; array[hole]= x; }

deleteMin with ‘Hole Trick’ The same ‘hole’ trick used in insertion can be used here too 2 5 4 3 6 2. Compare children and tmp bubble down if necessary 2 5 4 3 6 1. create hole tmp = 6 (last element) 2 5 3 4 6 3. Continue step 2 until reaches lowest level 2 5 3 4 6 4. Fill the hole

deleteMin PseudoCode void deleteMin() { if (isEmpty()) throw UnderflowException(); //copy the last number to the root, decrease array size by 1 array[1] = array[currentSize--] percolateDown(1); //percolateDown from root } void percolateDown(int hole) //int hole is the root position int child; Comparable tmp = array[hole]; //create a hole at root for( ; hold*2 <= currentSize; hole=child){ //identify child position child = hole*2; //compare left and right child, select the smaller one if (child != currentSize && array[child+1] <array[child] child++; if(array[child]<tmp) //compare the smaller child with tmp array[hole] = array[child]; //bubble down if child is smaller else break; //bubble stops movement array[hole] = tmp; //fill the hole

Heap is an efficient structure Array implementation ‘hole’ trick Access is done ‘bit-wise’, shift, bit+1, …

Heapsort (1) Build a binary heap of N elements the minimum element is at the top of the heap (2)   Perform N DeleteMin operations the elements are extracted in sorted order (3) Record these elements in a second array and then copy the array back

Build Heap Input: N elements Output: A heap with heap-order property Method 1: obviously, N successive insertions Complexity: O(NlogN) worst case

Heapsort – Running Time Analysis (1) Build a binary heap of N elements repeatedly insert N elements  O(N log N) time (there is a more efficient way, check textbook p223 if interested) (2) Perform N DeleteMin operations Each DeleteMin operation takes O(log N)  O(N log N) (3) Record these elements in a second array and then copy the array back O(N) Total time complexity: O(N log N) Memory requirement: uses an extra array, O(N)

Heapsort: in-place, no extra storage Observation: after each deleteMin, the size of heap shrinks by 1 We can use the last cell just freed up to store the element that was just deleted  after the last deleteMin, the array will contain the elements in decreasing sorted order To sort the elements in the decreasing order, use a min heap To sort the elements in the increasing order, use a max heap the parent has a larger element than the child

Sort in increasing order: use max heap Delete 97

Delete 16 Delete 14 Delete 10 Delete 9 Delete 8

One possible Heap ADT Template <typename Comparable> class BinaryHeap { public: BinaryHeap(int capacity=100); explicit BinaryHeap(const vector<comparable> &items); bool isEmpty() const; void insert(const Comparable &x); void deleteMin(); void deleteMin(Comparable &minItem); void makeEmpty(); private: int currentSize; //number of elements in heap vector<Comparable> array; //the heap array void buildHeap(); void percolateDown(int hole); }

Priority Queue: Motivating Example 3 jobs have been submitted to a printer in the order A, B, C. Sizes: Job A – 100 pages Job B – 10 pages Job C -- 1 page Average waiting time with FIFO service: (100+110+111) / 3 = 107 time units Average waiting time for shortest-job-first service: (1+11+111) / 3 = 41 time units A queue be capable to insert and deletemin? Priority Queue

Priority Queue Priority queue is a data structure which allows at least two operations insert deleteMin: finds, returns and removes the minimum elements in the priority queue Applications: external sorting, greedy algorithms deleteMin insert Priority Queue

Possible Implementations Linked list Insert in O(1) Find the minimum element in O(n), thus deleteMin is O(n) Binary search tree (AVL tree, to be covered later) Insert in O(log n) Delete in O(log n) Search tree is an overkill as it does many other operations Eerr, neither fit quite well…

It’s a heap!!!